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The Shooting Method and Nonhomogeneous Multipoint BVPs of Second-Order ODE

Boundary Value Problems volume 2007, Article number: 064012 (2007) Cite this article

Abstract

In a recent paper, Sun et al. (2007) studied the existence of positive solutions of nonhomogeneous multipoint boundary value problems for a second-order differential equation. It is the purpose of this paper to show that the shooting method approach proposed in the recent paper by the first author can be extended to treat this more general problem.

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References

  1. 1.

    Kwong MK: The shooting method and multiple solutions of two/multi-point BVPs of second-order ODE. Electronic Journal of Qualitative Theory of Differential Equations 2006,2006(6):1–14.

    Google Scholar 

  2. 2.

    Krasnosel'skiĭ MA: Positive Solutions of Operator Equations. P. Noordhoff, Groningen, The Netherlands; 1964:381.

    Google Scholar 

  3. 3.

    Baxley JV, Haywood LJ: Nonlinear boundary value problems with multiple solutions. Nonlinear Analysis: Theory, Methods & Applications 2001,47(2):1187–1198. 10.1016/S0362-546X(01)00257-7

    MATH  MathSciNet  Article  Google Scholar 

  4. 4.

    Il'in VA, Moiseev EL: Nonlocal boundary-value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Journal of Differential Equations 1987,23(7):803–810.

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Il'in VA, Moiseev EL: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Journal of Differential Equations 1987, 23: 979–987.

    MATH  Google Scholar 

  6. 6.

    Gupta CP: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. Journal of Mathematical Analysis and Applications 1992,168(2):540–551. 10.1016/0022-247X(92)90179-H

    MATH  MathSciNet  Article  Google Scholar 

  7. 7.

    Gupta CP: A note on a second order three-point boundary value problem. Journal of Mathematical Analysis and Applications 1994,186(1):277–281. 10.1006/jmaa.1994.1299

    MATH  MathSciNet  Article  Google Scholar 

  8. 8.

    Marano SA: A remark on a second-order three-point boundary value problem. Journal of Mathematical Analysis and Applications 1994,183(3):518–522. 10.1006/jmaa.1994.1158

    MATH  MathSciNet  Article  Google Scholar 

  9. 9.

    Constantin A: On a two-point boundary value problem. Journal of Mathematical Analysis and Applications 1995,193(1):318–328. 10.1006/jmaa.1995.1238

    MATH  MathSciNet  Article  Google Scholar 

  10. 10.

    Avery R: Existence of multiple positive solutions to a conjugate boundary value problem. Mathematical Sciences Research Hot-Line 1998,2(1):1–6.

    MATH  MathSciNet  Google Scholar 

  11. 11.

    Henderson J, Thompson HB: Multiple symmetric positive solutions for a second order boundary value problem. Proceedings of the American Mathematical Society 2000,128(8):2373–2379. 10.1090/S0002-9939-00-05644-6

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Ma R: Positive solutions for a nonlinear three-point boundary-value problem. Electronic Journal of Differential Equations 1999,1999(34):1–8.

    Google Scholar 

  13. 13.

    Sun W, Chen S, Zhang Q, Wang C: Existence of positive solutions to-point nonhomogeneous boundary value problem. Journal of Mathematical Analysis and Applications 2007,330(1):612–621. 10.1016/j.jmaa.2006.08.022

    MATH  MathSciNet  Article  Google Scholar 

  14. 14.

    Ma R: Existence theorems for a second order three-point boundary value problem. Journal of Mathematical Analysis and Applications 1997,212(2):430–442. 10.1006/jmaa.1997.5515

    MATH  MathSciNet  Article  Google Scholar 

  15. 15.

    Coffman CV, Wong JSW: Oscillation and nonoscillation of solutions of generalized Emden-Fowler equations. Transactions of the American Mathematical Society 1972, 167: 399–434.

    MATH  MathSciNet  Article  Google Scholar 

  16. 16.

    Raffoul YN: Positive solutions of three-point nonlinear second order boundary value problem. Electronic Journal of Qualitative Theory of Differential Equations 2002,2002(5):1–11.

    Article  Google Scholar 

  17. 17.

    Liu B: Positive solutions of a nonlinear three-point boundary value problem. Computers & Mathematics with Applications 2002,44(1–2):201–211. 10.1016/S0898-1221(02)00141-4

    MATH  MathSciNet  Article  Google Scholar 

  18. 18.

    Liu B: Positive solutions of a nonlinear three-point boundary value problem. Applied Mathematics and Computation 2002,132(1):11–28. 10.1016/S0096-3003(02)00341-7

    MATH  MathSciNet  Article  Google Scholar 

  19. 19.

    Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.

    Google Scholar 

  20. 20.

    Ma R: Existence theorems for a second order-point boundary value problem. Journal of Mathematical Analysis and Applications 1997,211(2):545–555. 10.1006/jmaa.1997.5416

    MATH  MathSciNet  Article  Google Scholar 

  21. 21.

    Mawhin J: Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics. Volume 40. American Mathematical Society, Providence, RI, USA; 1979:v+122.

    Google Scholar 

  22. 22.

    Mawhin J: Topological degree and boundary value problems for nonlinear differential equations. In Topological Methods for Ordinary Differential Equations, Lecture Notes in Mathematics. Volume 1537. Edited by: Furi M, Zecca P. Springer, Berlin, Germany; 1993:74–142. 10.1007/BFb0085076

    Google Scholar 

  23. 23.

    Gupta CP, Ntouyas SK, Tsamatos PCh: Existence results for-point boundary value problems. Differential Equations and Dynamical Systems 1994,2(4):289–298.

    MATH  MathSciNet  Google Scholar 

  24. 24.

    Gupta CP, Ntouyas SK, Tsamatos PCh: On an-point boundary-value problem for second-order ordinary differential equations. Nonlinear Analysis: Theory, Methods & Applications 1994,23(11):1427–1436. 10.1016/0362-546X(94)90137-6

    MATH  MathSciNet  Article  Google Scholar 

  25. 25.

    Gupta CP, Trofimchuk SI: Existence of a solution of a three-point boundary value problem and the spectral radius of a related linear operator. Nonlinear Analysis: Theory, Methods & Applications 1998,34(4):489–507. 10.1016/S0362-546X(97)00584-1

    MATH  MathSciNet  Article  Google Scholar 

  26. 26.

    Guo YP, Shan WR, Ge WG: Positive solutions for second-order-point boundary value problems. Journal of Computational and Applied Mathematics 2003,151(2):415–424. 10.1016/S0377-0427(02)00739-2

    MATH  MathSciNet  Article  Google Scholar 

  27. 27.

    Gupta CP, Trofimchuk SI: A sharper condition for the solvability of a three-point second order boundary value problem. Journal of Mathematical Analysis and Applications 1997,205(2):586–597. 10.1006/jmaa.1997.5252

    MATH  MathSciNet  Article  Google Scholar 

  28. 28.

    Gupta CP: A generalized multi-point boundary value problem for second order ordinary differential equations. Applied Mathematics and Computation 1998,89(1–3):133–146.

    MATH  MathSciNet  Article  Google Scholar 

  29. 29.

    Feng W, Webb JRL: Solvability of three point boundary value problems at resonance. Nonlinear Analysis: Theory, Methods & Applications 1997,30(6):3227–3238. 10.1016/S0362-546X(96)00118-6

    MATH  MathSciNet  Article  Google Scholar 

  30. 30.

    Feng W, Webb JRL: Solvability of-point boundary value problems with nonlinear growth. Journal of Mathematical Analysis and Applications 1997,212(2):467–480. 10.1006/jmaa.1997.5520

    MATH  MathSciNet  Article  Google Scholar 

  31. 31.

    Feng W: On an-point boundary value problem. Nonlinear Analysis: Theory, Methods & Applications 1997,30(8):5369–5374. 10.1016/S0362-546X(97)00360-X

    MATH  MathSciNet  Article  Google Scholar 

  32. 32.

    Cheung W-S, Ren J: Twin positive solutions for quasi-linear multi-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2005,62(1):167–177. 10.1016/j.na.2005.03.018

    MATH  MathSciNet  Article  Google Scholar 

  33. 33.

    Naito Y, Tanaka S: On the existence of multiple solutions of the boundary value problem for nonlinear second-order differential equations. Nonlinear Analysis: Theory, Methods & Applications 2004,56(6):919–935. 10.1016/j.na.2003.10.020

    MATH  MathSciNet  Article  Google Scholar 

  34. 34.

    Kong Q: Existence and nonexistence of solutions of second-order nonlinear boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2007,66(11):2635–2651. 10.1016/j.na.2006.03.045

    MATH  MathSciNet  Article  Google Scholar 

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Author information

Affiliations

  1. Lucent Technologies Inc., Lisle, IL, 60532, USA

    Man Kam Kwong

  2. Department of Mathematics, Statistics, Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045, USA

    Man Kam Kwong

  3. Department of Mathematics, The University of Hong Kong, Pokfulam, Hong Kong

    James SW Wong

  4. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    James SW Wong

  5. Chinney Investments Ltd., Hong Kong

    James SW Wong

Authors
  1. Man Kam Kwong
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  2. James SW Wong
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Correspondence to Man Kam Kwong.

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Kwong, M.K., Wong, J.S. The Shooting Method and Nonhomogeneous Multipoint BVPs of Second-Order ODE. Bound Value Probl 2007, 064012 (2007). https://doi.org/10.1155/2007/64012

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • General Problem